Integral Representations of Ratios of the Gauss Hypergeometric Functions with Parameters Shifted by Integers

Given real parameters a,b,c and integer shifts n1,n2,m, we consider the ratio R(z)=2F1(a+n1,b+n2;c+m;z)/2F1(a,b;c;z) of the Gauss hypergeometric functions. We find a formula for ImR(x±i0) with x>1 in terms of real hypergeometric polynomial P, beta density and the absolute value of the Gauss hyper...

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Vydané v:	Mathematics (Basel) Ročník 10; číslo 20; s. 3903
Hlavní autori:	Dyachenko, Alexander, Karp, Dmitrii
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Basel MDPI AG 01.10.2022
Predmet:	Asymptotic properties gauss continued fraction gauss hypergeometric function Gaussian processes Geometrical models Hypergeometric functions Integer programming integral representation Mathematical analysis Mathematical research Parameter estimation Parameters Polynomials Random variables Ratio and proportion Ratios Representations Israel
ISSN:	2227-7390, 2227-7390
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Abstract	Given real parameters a,b,c and integer shifts n1,n2,m, we consider the ratio R(z)=2F1(a+n1,b+n2;c+m;z)/2F1(a,b;c;z) of the Gauss hypergeometric functions. We find a formula for ImR(x±i0) with x>1 in terms of real hypergeometric polynomial P, beta density and the absolute value of the Gauss hypergeometric function. This allows us to construct explicit integral representations for R when the asymptotic behaviour at unity is mild and the denominator does not vanish. The results are illustrated with a large number of examples.
AbstractList	Given real parameters a,b,c and integer shifts n1,n2,m, we consider the ratio R(z)=2F1(a+n1,b+n2;c+m;z)/2F1(a,b;c;z) of the Gauss hypergeometric functions. We find a formula for ImR(x±i0) with x>1 in terms of real hypergeometric polynomial P, beta density and the absolute value of the Gauss hypergeometric function. This allows us to construct explicit integral representations for R when the asymptotic behaviour at unity is mild and the denominator does not vanish. The results are illustrated with a large number of examples. Given real parameters a,b,c and integer shifts n[sub.1] ,n[sub.2] ,m, we consider the ratio R(z)=[sub.2] F[sub.1] (a+n[sub.1] ,b+n[sub.2] ;c+m;z)/[sub.2] F[sub.1] (a,b;c;z) of the Gauss hypergeometric functions. We find a formula for ImR(x±i0) with x>1 in terms of real hypergeometric polynomial P, beta density and the absolute value of the Gauss hypergeometric function. This allows us to construct explicit integral representations for R when the asymptotic behaviour at unity is mild and the denominator does not vanish. The results are illustrated with a large number of examples.
Audience	Academic
Author	Dyachenko, Alexander Karp, Dmitrii
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Snippet	Given real parameters a,b,c and integer shifts n1,n2,m, we consider the ratio R(z)=2F1(a+n1,b+n2;c+m;z)/2F1(a,b;c;z) of the Gauss hypergeometric functions. We... Given real parameters a,b,c and integer shifts n[sub.1] ,n[sub.2] ,m, we consider the ratio R(z)=[sub.2] F[sub.1] (a+n[sub.1] ,b+n[sub.2] ;c+m;z)/[sub.2]... Given real parameters a,b,c and integer shifts n1,n2,m , we consider the ratio R(z)=2F1(a+n1,b+n2;c+m;z)/2F1(a,b;c;z) of the Gauss hypergeometric functions. We...
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SubjectTerms	Asymptotic properties gauss continued fraction gauss hypergeometric function Gaussian processes Geometrical models Hypergeometric functions Integer programming integral representation Mathematical analysis Mathematical research Parameter estimation Parameters Polynomials Random variables Ratio and proportion Ratios Representations
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Title	Integral Representations of Ratios of the Gauss Hypergeometric Functions with Parameters Shifted by Integers
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